Kostka polynomials and affine Kac-Moody algebras

PI建议研究表示理论和组合数学，根据仿射（更一般地可对称化）Kac-Moody代数的Hall-Littlewood和Kostka-Foulkes多项式的某些推广。具体来说，PI希望从Cherednik-Macdonald理论，q-超几何级数和正性的角度研究仿射Kostka-Foulkes多项式。PI还希望使用对应于任意Kac-Moody代数的多项式来更清楚地说明这些知之甚少的李代数的表示。对称函数是多元多项式，当变量被置换时保持不变。他们承认一个丰富的理论，涉及代数，组合学和表示论的相互作用。 Hall-Littlewood多项式是对称函数，其依赖于 额外参数和插值之间的两个非常重要的对称函数类。霍尔-利特尔伍德多项式的理论可以追溯到霍尔、利特尔伍德和后来的麦克唐纳的作品。这些多项式自然发生在不同的数学领域，包括代数，几何，表示论，组合数学和数学物理。它们有许多重要的性质和应用，我们的目标是研究经典的Hall-Littlewood多项式到无限维（Kac-Moody）李代数的推广，并在此基础上推导出经典性质的类似物。PI希望一方面澄清Hall-Littlewood多项式的推广与它所涉及的经典对象（如“双仿射Hecke代数”）的推广之间的关系。

The PI proposes to study the representation theory and combinatoricsunderlying certain generalizations of Hall-Littlewood and Kostka-Foulkes polynomials for affine (and more generally symmetrizable) Kac-Moody algebras. Specifically, the PI wishes to study affine Kostka-Foulkes polynomials from the perspectives of Cherednik-Macdonald theory, q-hypergeometric series and positivity. The PI also hopes to use the polynomials corresponding to arbitrary Kac-Moody algebras toshed more light on representations of these poorly understood Lie algebras.Symmetric functions are multivariate polynomials which remain unchangedwhen the variables are permuted. They admit a rich theory involving an interplay of algebra, combinatorics and representation theory. Hall-Littlewood polynomials are symmetric functions that depend on an extra parameter and interpolate between two very important classes of symmetric functions. The theory of Hall-Littlewood polynomials traces its origins to works of Hall, Littlewood and later of Macdonald. These polynomials occur naturally in diverse areas of mathematics includingalgebra, geometry, representation theory, combinatorics and mathematical physics. They have many important properties and applications.The goal of the proposed research is to study the generalization of the classical Hall-Littlewood polynomials to the case of infinite dimensional (Kac-Moody) Lie algebras and to derive analogs of classical properties in this general setting. The PI hopes to clarify the relation between the generalizations of Hall-Littlewood polynomials on one hand and the generalizations of the classical objects it is related to (such as the "double affine Hecke algebra").